Straight-line course computer



Jan. 30, 1951 J. B. GEHMAN STRAIGHT-LINE COURSE COMPUTER 2 Sheets-Sheet 1 Filed March 5l, 1950' afd-sim L 0 m w INVEN-ron @Ivn B. emazz ATTOINEY 2mm @92X Jan. 30, 1951 J. B. GEHMAN STRAIGHT-LINE COURSE COMFUTER Filed March 51', 1950 2 Sheets-Sheet 2 ATTORNEY` Patented Jan. 30, 1951 salmion'r Li-Nn` CoUItsE COMPUTER John Gehman, iiaddonfield, .liiassignor to Badio Corporation of America, a corporationof Delaware Application March 31, 1950, Serial No. 153,150

This invention relates generally to navigation systems and more specically to a system for guiding a vehicle along a straight line' path as referred to two iixed positions.- l

Occasions arise wherein a vehicle such as an airplane is required to travel in a straightline patlr with a predetermined heading angle and ati a; predetermined ground speed repetitivelfy. In aerial photography, or patrol-or Search work,vr the paths traveled are parallel straight lines which resemble a grid.

It i'sfan objfec't of thefpr'eseiit invention to p`r0-` videfa computer which supplies data for guiding a vehicle along a. straight line ground path;- with aipredetermined direction of ground motion and ata predetermined ground speed.

It isy another. object of they present invention toiprovide a: computer which automatically solves for the length` of one side in eachy of two adjacent triangles whenl provided with datav as to the lengths' of Athe other? two sides' in eachtriangle and the included angles.

These andA other-objects are achieved" in the presentfi'nven-t'ionby a` computer which automatically' determines thei distances aA vehicle should be from` two iixedV points in accordance with its:'predeterminedvspeed and heading angle;

8` claims. (ci. 235-61) These" distancesl are expressed as" potentials. o

'These distances are con'npared withv thev actual 'distance from these. .two points lwhich are also expressed as:v potentials.- Anyerror voltages whichoccur as aresult of-.these comparisonsA-may beusedtoactu-ate an indicator or to automatically correct the course and speed: ofthe'f vehiele'to compensate forfany d'e'viati'onsithereof-l The -two: iixed'positiel'nsV relier-red' to.y herein are twovstationsiwhich areat'fabl-r-nown distancel'apant andi have radio" transmitting apparatus of .the type which responds tolsignals transmittedfiroin theairplane to 'supply' information" aste thesaur-` plane-ts f actual.` distance'l from these two points. Apparatus for" this-H4 type 'ofi navigation systeni is descrled and 'claimed an application byl S. W. Seeley,.SerialiNo.` 638,3373-now Patent'Noa 2,5126,- 281; iiledx. December 29, 1945,. and assigned toia common"as'signeey The distance between-theit'wo pointsis' known;` The `direction of groundmo tiOnnr'heading; o'f the vehicle orangle it makes with: the-base linefor-nied'by th'e line joining.: Ythe TheS tV/' cl'o'inllitd diStanCS flX th'ldt'i of the Vehicle iiaccordance with the prdtr'nind headingand speedan'd they` are coin-pared with the actual distances from the iixe'depoints to determine if any differences exist.

The novel features of the' invention as well as the invention itself, both as to its organization and rretl'iodl of operation, wil-1 bestbe understood from-the following description,- when read in con- Iectn with the accompanying drawings; in Which:

Figure l is a g'raphillustratirrg the problem to be solved, Y

Figure 2 is? a` schematic diagram of abasto syslten finding application in an embodiment of the present invention for obtaining a potential pro portinal to the product of two functionseii; pressed2 as shaft positions. 1

Figure 3- is"` a schematicdiagr'am ofV one embodiment or the invention,-

Figure 1i is a circuitV diagram illustrating one type ofuneterffr displaying4 the inform vided by the embodiment of thefinven-t-ion;

,Figures 5, 6 and '7 are' graphsilllstratinig the effects'on the error voltagesof a-'number of possible deviations from the predetermined heading' entls'peed;` i

Figure 8 isl a circuitv diagram( of apparatus for deriving?speedVv and turn error voltagesffron com# parison-offthe computed distance voltages with theA actualdistance voltages,-` 1

Figure'Q- i's a circ'li't dia"gram: ofVv anotherbasio system which iinds application in another ern'A bodiment of the present invention,

Figure 10 is a circuit diagram' of a low pass filter, and

Figure 11 is aschematic, diagram of another embodiment" of'ythe invention;

Referring'toFiguiie'l, points A'a'nd' B- are'r'adi'o beacon stations'of thetyiJe'described the abve identified applicatioriffy SeeleyV and are'liiwnfas Shora stations. The dstaiicefbet'wen 'a'nd B is accurately known. Y Point uM representstile'k airplane or vehicle" position. The airplane must' travel along a strai'ghtline course at'a heading angle 0'; Thisheading angle@ isfthe anglernade bytheairplane c'oursewiththe bfafs'e line 'extend-f ing through'pointsA and B; The distances vda and rifare*V the respective distances between the' point X1, whichis'the intersection'of' the course with the base line, and points" A aif''dBi These distances da and di are known. The distance'"be-A tween'M' andiX representstheproduct"of the desired speed of"travel and' the tiin'eV elapsed since leaving the base line'and'isiridicated as" p.' ifa plane is to fly at a given heading angle'in a straight line 'at apredetermi'ned speed,A thep'r'ob lem il;"to"continuusly`V solve; for the" distances' 'the plane should be" from both xedpoints inl 35;" cordance with those predetermined factorsA Comparison can be continuously made beiW aesaic the computed distances and the actual distance information supplied by beacon points A and B for the purpose of compensating for any differ ences.

It will be readily recognized, that two adjacent triangles are shown in Figure 1 and in each triangle two of the sides and the included angle are known. The third sides, Arepresenting the distances from the plane to the xed points A and B may be readily found from the equations:

Referring to Figure 2, there may be seen a schematic representation of an impedance bridge l having a pair of input terminals I2 to which a potential is applied and a pair of null detecting terminals I4 for detecting a balance. The bridge has four impedance arms I6, i8, 20, 22 or two pairs of opposed impedance arms. Two functions f1 and f2 are expressed as the positions of two shafts. The impedances in one of the pairs of opposed arms I 6, i8 (1) are variable, (2) are respectively calibrated in terms of f1 and f2 and (3) are respectively coupled to be varied by the two shafts. The impedance of one of the other of the pairs of opposed arms is a constant C. The impedance in the remaining arm 22 is variable H and has a value selected to permit the balancing of the 'bridge for all impedance variations caused in the variable arms. A detector 24 is coupled to the null detecting terminals i4 and its output is coupled to a servomotor 26. The servomotor shaft 28 is mechanically coupled to the variable impedance arm 22. The mechanical coupling is represented by a dotted line. The servomotor shaft 28 is also coupled to the mov able arm 3D of a potentiometer 32.

It will be readily apparent that the bridge il?, detector 24, and servomotor 2 form a servo loop wherein the servomotor always turns in a direction to reduce the voltage applied to its terminals by the detector to a minimum. In View of the mechanical coupling to the H impedance arm of the bridge, for any setting of the arms f1 and f2, the servomotor will adjust arm H until the bridge is balanced.

For bridge balance,

Therefore, for any shaft settings of the arms f1 and f2, the servomotor shaft turns an amount proportional to the product of the shaft settings. Accordingly, the potentiometer movable arm 32 which is coupled to the servo shaft is moved an amount proportional to the product of the two functions. The output voltage from the potentiometer is also proportional to the product of the two functions. If f1 and f2 are always equal, these two arms may have their variable impedances ganged' to be movable simultaneously. The voltage output of the potentiometer, accordingly, is proportional to the square of the function which is applied to the bridge as a shaft position.

Referring now to Figure 3, there is shown a schematic diagram of an embodiment of the in` vention for computing d12 and (Z22 in accordance with the computed course for the plane. A motor 3B is controlled in speed and direction of shaft rotation by any well known motor speed and direction control apparatus 34. The motor shaft is coupled to drive two gangs-d, opposed, variable impedance arms of a rst bridge 38 which are `'calibrated in terms of p. The motor speed is controlled so that its shaft changes the impedance p at a rate in proportion to the predetermined speed of the plane for the course chosen. The first bridge, detector 4B, servomotor 42 and potentiometer 44 are similar to those shown and explained in Figure 2. The motor is started as soon as the plane crosses the base line between points A and B. Accordingly, the value of p applied to the bridge 38 is a product of the predetermined speed and the time elapsed since the plane intersected the base line. The shaft of the servomotor, accordingly, is turned an amount proportional to p2 and the output potential of the potentiometer 44 is proportional to p2.

A second bridge 46 is provided having a pair of ganged, movable arms calibrated in terms of the distance d3. The distance d3 is known for the course chosen and is set into the bridge. The second bridge 46, detector 48, servomotor 50, and potentiometer 52 function to provide a potential proportional to (232 or to the square of the distance between point A and the point of intersection of the vehicle with the base line function in the manner as described for the system shown in Figure 2.

A third bridge 54 is provided in which one variable impedance arm is calibrated in terms of the distance p and is coupled to be driven by the motor shaft. An opposed variable impedance arm is calibrated in terms of the distance da and is mechanically coupled to be driven by the shaft that is used to set in the known value da. Accordingly, the potentiometer movable arm 62 which is driven by the servomotor 58 of this bridge 54 is positioned proportionally to the product of ,o and d3.

' A center-tapped potentiometer E4 is connected across a source of potential and is calibrated so that its movable arm may be adjusted to provide an output potential proportional to -2 cos 9. Since 6, the heading angle, varies in value between zero and 18) degrees (ccs 0 varies between zero and plus or minus one), a potentiometer such, as the one shown herein 64, which has its re-- sistance winding tapped at the center andcon-` nected to the midpoint of a potential source to: which the outer ends of the winding are con. nected, may readily be calibrated to provide an output proportional to -2 cos 0.

The p2 potentiometer 44 movable arm is connected to one side of the 132 potentiometer 52. The potential at the 1132 potentiometer arm is then representative of p2+d32- The movable arm of the daz potentiometer 52 is then connected to one side of the pds potentiometer 6B. The output of the -2 cos 0 potentiometer 64 is applied to the input of the pds potentiometer $9. Therefore, the potential between the pda potentiometer movable arm G2 and ground is representative of p2|d322pds cos 0 which is equal to 312 computed. A solution for d?! in the adjacent triangle (see Fig. 1) is similarly found. Another potentiometer 66 is ganged to the shaft of the servomotor 42 to provide an output potential proportional to p2. A fourth bridge, 68, detector l0, servomotor l2 and potentiometer 14 are provided for obtaining a potential proportional to d42. The distance d4 is a known value, being the distance from the point of intersection of the plane with the base line to the point B.

The shafts which represent p and d4 are also coupled to two opposed variable impedance arms of a fifth `impedance bridge 16 which is also t the ser'ioshaftougtput, is. accordingly positipped proportionally to the producty pdi. lThe latter potentiometer 82 has applied to it the out?A put of a center-tapped potentiometer 86 whose movable arm is positioned in accordance with L2 cos (180-0). The potentiometers 66, 14, 82, 86 are connected to each other, as expiained above .formalism-ine di?, in. a fashion` to, add their outputs The. netutial between, the par potentiometer movable arm 84 and ground then represents p-Jrdig-Zpdi-cos (180'4-0) which equals daz computed. All the bridges shown in Figure 3 have. .the terminals,d tot which potential isapplied, brought. out andshown as arrows. A sou-ree ofi rpotential and connections from the bridges to this source are not shown, since they would .onlyv serve to. complicate the drawing.

Figure L is a. schematic diagram ofa meter which is used to show a 4deviation between the actual and the computed position of the plane. The radio navigation apparatus described in the above `noted Seeley application provides the actual distancesl between the plane and the two fixed points-in the form of shaft positions. These shafts may be coupled to two bridge type servo loops as described and shown in Figure 2 and voltages proportional to dizr actual' and diz actual maybe obtained from potentiometers ganged to the respective servomotor shafts. The actual and computed'V di2 are opposing'ly applied' to one winding 900i? thecrossedpointermeter 88 shown in Figure 41 The actual and computedY dzz areV opposingly applied1to the other winding 92 of thev crossed,poifntermeterf 8-81 When the crossed pointers read zero, the plane isfollowing" its predeterminedicoursef.

y Figures 5, 6- andf 7= respectively represent the conditions when the heading ang-le is approaching Zerodegrees, 90=1degrees`and 180r degrees. Referring to Figurev 5, i-,fM the plane ltravels faster than: its predetermined speed, the distanceA d2 actual is greater thant the distance d2 computed. It the plane travels slower than its computed speed, d2 actual is less than d2 computed. For .eitherf the. slow on the- 4fast speed-l cond-ition of the` planev the difference between di actual and dni computedis negligible anddoesnot appear. If .theplanesiss toithex right of its computed; course, di. actual` is. greaterl than d1. computed.l Ifthe plana is. to the. left off its computed coursev d1 actual. is. less than. al1Y computed. For eitherV the rightron left condition.thedifference between d;

actuah anddacomputedisfnegligible' andddoes not appear.; The above indicated conditions are set fprthimtherformoffa table with a -|1`-signf in;- dicating. that the actualudista-nce is-greatenthan the, computed' distanceY and; a fe-*t sign indica-t'- ingthat the. actuaL distance is less Vthan the corn;- putedidistancec alla. dzsca.

. 'col-F,

ForJ thef 9 0:- deg-ree heading angle condition, as, shown inFig-ure 6'; theta-ble appearsas follows:

'diam asaaeis For the situation shown -in 'Figure *7i where .the heading angleapproaches 1&0.l degrees., the table appears as follows:

1213.116` dict. 4v12net.

Too Fast -l- 0Y 'loolowl hll 0i TQ Right n 0 To Leit- 0 For manual operation of the plane, the meter shown in Figurel 4 may be observed and correction in the plane-s-. speed and directionl `may Abe made until the crossed pointers are both broughtA back to a zero reading. 'By substituting A.C. generators which are synchronized, or a single lA.C; source for all the potentiometers, A.C. lvoltages proportional'L to diz computed and' Z22 computed are obtained. These may be compared with A.Cz voltages proportional to dll' actual .and d2? -actual andI the resultant error voltages used for automaticcontrol: of the plane.

In Figure 8 a diierential resolver' 94 shown schematically. This. resolver 94 hasra stator with two stator windings. 96; 98a Each of theseA windings is center tapped and the windingsy are oriented 'at right angles'. with respect to. one an; other; The eld whichthey. setupr whenexcit'ed, is; the;v vector addition of the field from. each winding.4 The rotor.' ofthe. angle resolver also has` two. separate windings |00,` |02 disposed to be; rotated. within the fieldsv setupA byz the stator windings,A The rotor windings |00; |02.v are ori.- enteol atright angles; with respect to each other. The field which theyset' up-,.if1 they; were: to: be: excited, would befthe vectorv addition. of theiiel'd finmpeachfwinding.

The voltages diz: actual; and d12` computed are n opposingly connected `tot one stator: winding 96 and the voltage di? actual' and d22 computed'. are QQIUQSngy Connetedl to ther.` other stator winding 88. The rotor shaft|04 is displaced fromy areferxence position atV an` angle;v corresponding tol the headinga-ngle. The. rotor shaft reference position (6=0) should:v be,l the` one. at which each' of the rotor-coils |00, |02 is parallel; to each of. the stator coils 96, 918 or is in amaximurn inducing relationship withgit: The,rotor-'winding` l |32r parallel to' the statorhwinding 96 to which divoltfages are appliedfistheawinding which produces an error voltage` output representative. of` an error in direction; Therotor winding |00, parallel to the stator winding 98' to which d22-voltages are applied, is the: winding which produces'v an error .voltage output representativey oft` an4 error in speed. TheA angle of" rotor. shaftl rotation is such that Vitgiseolual tofone-half the heading angle. Operation ofthe differential resolver'thus occurs in v degrees,ofrevolutionz Any quadrant may be selected providing due care is taken in observing the proper: polarity in connecting the rotor Windingsto operate"automaticacontrol equipment'.

From the previously" noted tables it can be Vseen;` that the;- appearance. oft anA error signal` as well as its relative" value varieswith the heading angle. The output of the angle resolver, when operated on by the angle 0, shows consistent results for speed and turn indications and may be directly connected to the planes automatic control equipment. When a combination' of an error in speed and being either to the right or to the left of the course occurs, either both correo.- tions= aremade simultaneously or, if,v in' view of the errorinthe planes position one error/voltagev 7 erate to correct 'the planes course until the error voltage which was cancelled begins to appear because of the correction which has occurred.

In Figure 1 a number of parallel paths for an assumed mapping operation are shown as dotted lines. The heading angle and speed may remain the same for all of them. Only the distances dxY and d4 change for each path. When the plane has flown to the end of its run, the shaft positions for d3 and d4 are set to their new values and-the motor turning the shaft at the predetermined speed is stopped. The plane is then turned and brought to the next path. When the d12 and d22 voltages, which are computed with the new values of da and d4, equal the actual di2 .and 1122 values, the motor is started in a reverse direction and the plane may be automatically controlled again. This procedure may be followed at each end of the course. Upon reaching the base line the motor is again stopped and the new values of da and d4 are set into the computer.

The plane is turned to the new position on the base line as located from the two beacon stations andv the motor is started in a direction to increase p.

- For the situation where the selected course for -the plane is on both sides of the base line, the computer is also readily adaptable for use. All that is required is a reversing switch that automatically reverses the direction of rotation of the motor which-produces the shaft position p when the base line is crossed (when p equals zero). There is also required a mechanism to switch the setting of the -2 cos 0 potentiometer to -2 cos (1800) and the setting of the -2 cos (180-6) potentiometer to -2 cos 0. The potentiometers may be switched by providing auxiliary potentiometers for the -2 cos 6 and -2 cos (l80-0) potentiometers in the computer and switching the potentiometers in and out of the circuit by means of relays also actuated by the switch which reverses the motor. The differential resolver may be manually turned to the complement of the heading angle or an auxiliary motor operated by relays may be provided to perform this operation.

Referring now to Figure 9, there is shown a circuit diagram ofanother means for obtaining a potential proportional to the product of two functions expressed as positions of a shaft. The circuit shows a variable frequency oscillation generator ISS of the well-known type described in Patent No. 2,268,872 by William R. Hewlett. In this type of oscillator the frequency of oscillation is determined by the values of the variable, series-connected, resistor |98 and condenser Hll combination connected in series with a variable, parallel connected, resistor H2 and condenser -I I4. The first electron tube I |6in the oscillator is an oscillation stage, the second electron tube H8 is a phase shift stage. The third electron tube |25 is a stage of amplification with a condenser |22 across its output.

It can vbe shown that the frequency of the oscillation generator is determined by Ganglng the variable resistors so that their values are always equal and gauging the variable con= densers so that their values are always equal leads to 1 1 f afec or fN Where R is the value of either resistor |08 or I I2 and C is the value of either capacitor HI) or H4.

Applying a series of high impedance and a shunt condenser to the output of the oscillation generator represented in Figure 9 provides or the potential across the output condenser is proportional to the product of the value of either resistor IGS or H2 and either condenser H0 or H4. The pentode amplification stage |20 provides the required high series impedance and the condenser |22 across the output is the shunt condenser. Therefore, if the ganged variable resistors |08, I I2 are moved to a position proportional to one function expressed as a shaft position and if the ganged variable capacitors H0, H4 are moved to a position proportional to a second function expressed as a shaft position, the potential existing across the output condenser |22 is proportional to the product of the two functions. If the two functions have the same values the output potential is proportional to the square of one of the functions.

Figure 11 represents another embodiment of this invention utilizing the circuit shown in Figure 9. A motor speed and direction control mechanism |28 and a motor |30 are used, in similar fashion as described for Figure 3, to generate a shaft position proportional to p. A rectangle identified as p2 generator |32 is representatve of the circuit shown in Figure 9. The motor shaft is coupled to the ganged variable resistors and ganged variable condensers of the p2 generator |32 to vary them simultaneously. These condensers and resistors are calibrated in terms of p instead of frequency. The output of the p2 generator |32 is a voltage proportional to p2 which is then rectified by a rectifier |34 to be a D. C. potential.

A dag generator |36 is provided which is also a circuit similar to the one shown in Figure 9. The variable condensers and resistors of the daz generator |36 are all ganged to be simultaneously variable and equal and are calibrated in terms of da. Therefore, when the value of da is set into the generator an output potential is provided proportional to daz. This is also rectified by the rectifier I38 represented schematically.

A third pds generator |40 is provided having its ganged, variable resistors coupled to the motor shaft to be positioned thereby and calibrated in terms of p. Its ganged, variable condensers are coupled to the da shaft to be positioned thereby and are also calibrated .in terms of da. The pda ...andthe deviation of said vehicle from Xtermined course.

generator |40 output is thenpds. This'outp'utiis "rectified by thefrectiiier |42 and applied across the winding of a center tapped potentiometer |44. Output from the potentiometer |44 is taken from its movable armand the center tap. The pot'entiometer is calibrated in terms of -2 cos 6 as described for the potentiometer in Figure 3. The potentiometer movable arm is then positioned'in accordance with the value of -2 cos 0 1 andthe output is then -2d3p cos 0. tial outputs of'the two rectiiiers |34, |38 and the potentiometerv |44 are then added in series and l their sum is a potential proportional to the value of diz."

The poten- In similar fashion, using three potential genfferators |46, |48,` |50 as shown in Figure 9, po- V'tentials 'proportional to p2, 0142, and pd4 are generated from the respective shaft positions. These u potentials are then rectified by the respective rectiers |52, |54, |56. Therectied potential proportional to pd4 is then applied to a center 4Atapped potentiometer |58 calibrated in terms of 'f- 2 cos (18W-0). The potentiometer |58 out- ''put',` and theyrectied potentialsrepresentative of pz and diz are then added together to provide at Jthe output terminals a potential proportional to 'd`22.

then be utilized in the above described fashion for guiding the vehicle. What has been herein described and shown is I a novel computer system which provides infor- The computed potentials di? and Z22 may mation for guiding a plane ralong a predetermined :straight line 'path at a predetermined ground speed by providing potentials which are propor- `Y tional to the square of the distances between the plane and two fixed points. These potentials are representative of the distance ,the vehicle should plane to the course from which it has deviated.

What is claimed is: Y l. In a computer for guiding a vehicle along a predetermined course at a predetermined speed in response to information based `on the position of said vehicle withrrespect to a rst and a lsecond point, the combination of means for pror ducing a first potential proportional to the square of the distance said vehicle should be from said first point according to said predetermined course and speed, means for producing a second potential proportional to the square of the distance lvsaid vehicle should be from said second point *according to said Hpredetermined course and speed, means for producing a third potential proportionalV to the square of the actual distance of sa-id vehicle from said rst point, means for producing a fourth potential proportional to the Isquare ofthe actual distance of said vehicle from jsaid second point, means for combining said rst and third potentials and said secondand fourth potentials to provide an indication'of the deviation between the actual and predetermined speed its prede- V.2. in a computer for'guiding a 'vehicle along a ,predetermined course at a lpredetermined speed in response to informaticnbased on the position of said vehicle with respect to' a first and a second point, theangle made by said course with the base tance between the point of intersection of said course with said base line and said two points, means for generating potentials proportional to the squares of each of two sides, the product of seid two sides and'twice the negative of the cosine of the angle included between said two sides, in each of the two triangles formed by the vehicle position according to said predetermined course and speed, said 51st and second points and said point of intersection of Vsaid vehicle course with said base line, one of said two sides in eachk triangle being the common side formed by the distance between said point of intersection and said vehicle position according to saidpredetermined ,course and speed, the other Vof said two sides in each triangle being the side formed by the portion of said base line included in the respective triangles, means to combine the potentials produced for each triangle `to provide for each triangle a potential'proportional to the square of the third side, means to produce foreach triangle a potential proportional to the square of the actual length ofsaid third side, and means to combinesaid third side potentials for each triangle to provide error potentials indicative of the deviation between the actual and predetermined speed and the deviation of said vehicle from said predetermined course.

3. In a computer for guiding a vehicle along a predetermined course at a predetermined speed in responseto information based on the position of said `vehicle with respect to a first and a sec- -ond point, the angle made by said course with the base line joining said first and second points and the distance between the point of intersection of said course with said base line, the instant at which said vehicle crosses said base line, and

said first and second points, the combination of i means for producing nine potentials as follows: Ya first potential proportional in value to the square of the product of said predetermined speed and the time elapsed since the vehicle left said point of intersection, a second potential proportional in Value to the square of the distance from said first point to said point of intersection, a third potential proportional in value to the product of the product of said predetermined speed and the time elapsed since the vvehicle leit saidpoint of intersectionV and said distance from said iirst point to said point of intersection, a fourth potential proportional to the negative of twice the cosine of the angle made by said vehiclecourse with that-part of the baseline between saidpoint of intersection and said first point, a fth potential proportional to the square of the distance between said point of intersection and said second point, a sixth potential proportional to the product of said distancebetween said point of interection and said second point and said productof said predetermined speed and the time elapsed since said vehicle left said point of intersection, a seventh potential proportional to the negative of twice the cosine of the angle madevbysaidfvehicle course with that part of the baseline betweenv said point of intersection and said second point, an eighth potential proportional to the square of actual distance between said ve- 'liicle and said first point, a ninth potential proportional to the square ofthe actual distance between said vehicle and said second point, means l l to the square of the computed distance between said vehicle and said first point, means to multiply said sixth and seventh potentials, means to add the product of said sixth and seventh potential to said first potential and said fifth potential to provide an eleventh potential proportional to the square of the computed distance between said vehicle and said second point, and means to combine said eighth and tenth potentials and said ninth and eleventh potentials to provide error potentials proportional to the difference between the actual and predetermined speed along said course and proportional to the distanceto the right or left of said course.

4. A computer as recited in claim 2 wherein each of said means to produce a potential proportional to the square of a distance comprises an impedance bridge having a rst and a second pair of opposed arms, a pair of input terminals and a pair of null detecting terminals, the impedances in each of said first pair of arms being calibrated in accordance with said distance, the impedance in one of said second pair of arms being a constant and the impedance in the other of said second pair of arms being adjustable and having a value selected to permit balancing of said bridge for impedance variations of said rst pair of arms, a detector coupled to said null detecting terminal, a servomotor electrically coupled to said detector, the shaft of said servomotor being coupled to the adjustable one of said second pair of arms, a potentiometer calibrated in terms of said distance to be squared, the movable arm of said potentiometer being coupled to be driven by y said servo motor shaft, each of said first pair of arms being adjusted proportional to a distance whereby said servomotor adjusts said adjustable arm of said second pair of arms to balance said bridge and said potentiometer movable arm is positioned to provide an output potential proportional to the square of said distance.

5. A computer as recited in claim 3 wherein each of said means to produce a potential proportional to the square of a distance comprises a variable frequency oscillation generator of the variable resistance-capacity type wherein the frequency determining elements consist of a series connected variable resistor and variable condenser 'connected in s-eres with a parallel connected variable resistor and variable condenser, said variable resistors being ganged to be simultaneously adjustable and to have equal values, said variable condensers being ganged to be simultaneously adjustable and to have equal values, said variable resistors and said variable l condensers each being calibrated in terms of said distance, a high impedance connected in series with the output from said oscillator, and a condenser connected in parallel with said oscillation output through said high impedance, the potential output across said condenser being proportional to the product of the distance to which said variable condensers are adjusted and the distance to which said variable resistors are adjusted.

vwith the base line joining said rst and second points, the instant at which said vehicle crosses said base line, and the distance between said point of intersection and said rst and second points expressed as shaft positions, mean to represent the product of said time elapsed since crossing said base line and said predetermined speed as a. product shaft position, means to generate a rst potential from said product shaft position proportional to the square of said product, means to generate a second potential from said shaft position representative of said distance from said intersection to said first point proportional to the square of said distance, means to generate a third potential from said product shaft position and said latter named shaft position proportional to the product of said product and said distance from said intersection to said first point, means to generate a fourth potential proportional to the negative of twice the cosine of the heading angle, means to generate a fifth potential proportional to the product of said third and fourth potentials, means to generate a sixth potential from said shaft position representative of the distance between said point of intersection and said second point proportional to the square of said distance, meansto generate a seventh potential from said product shaft position and said latter-named shaft position proportional to the product of said product and the distance between said point of intersection and said second point, means to generate an eighth potential proportional to the negative of twice the cosine of the supplemental angle of the heading angle, means to obtain a ninth potential proportional to the product of said seventh and eighth potentials, means to add said first, second and fth potentials to obtain a tenth potential proportional to the square of the distance of said vehicle from said rst point according to said predetermined speed and heading angle, means to add said first, sixth and ninth potentials to obtain an eleventh potential proportional to the square of the distance of said vehicle from said second point according to said predetermined sneed and heading angle, means to generate a twelfth potential proportional to the square of the actual distance of said vehicle from said first point, means to generate a thirteenth potential proportional to the square of the actual distance of said vehicle from said second point, means to combine opposingly said tenth and said twelfth potentials and said eleventh and said thirteenth potentials to obtain error potentials proportional to the deviation of said vehicle from said predetermined heading and from said predetermined speed.

'7. In a computer for guiding a vehicle along a predetermined course at a predetermined speed in response to information based on the actual position of said vehicle with respect to a first and a second point, the angle made by said course with the base line joining said rst and second points, the instantat which said vehicle crosses said base line, and the distance between said point of intersection and said rst and second points expressed as shaft positions, the combination of means for converting the product of the time elapsed since said vehicle crossed said base line and said predetermined speed into a representative shaft position, means for converting said latter named shaft rposition into a shaft position representative of the square of said product, means for converting said latter named shaft position into a first potential proportional to the square of said product, means for converting said shaft position representative of the distance between said point of intersection and said first point into a shaft position representative of the square of said latter named distance, means for converting said latter named -f point of intersection and said rst point, means for obtaining a shaft position representative of f the product of said predetermined speed and the time elapsed since said vehicle crossed said baseline and said distance from said intersection to said rst point, means for converting said latter named shaft position into a third potential proportional to the product of the product of said predetermined speed and the time elapsed since said vehicle crossed said base line and said distance between said point of intersection and said first point, means to produce a fourth potential proportional to the negative of twice the cosine of the angle made by the course of said vehicle with 'said baseline, means tor multiply said third and fourth potentials to obtain a fifth potential representative of said product, means to convert said shaft position representative of therdistance between said point of intersection and said second point into a shaft position representative of the square of said latter named distance, means to c-onvert said latter named shaft position into a sixth potential representative of the square of said distance between said point of intersection and said second point, means to convert into a shaft position representative ,of their product the shaft positions representative of the product of the time elapsed since said vehicle crossed said base line and said predetermined speed, and said distance between said point of intersection of said second point, means torconvert said latter named shaft position into a seventh potential representative of said latter named. product, means to produce an eighth potential proportional to the negative of twice the cosine of the angle included between said heading and that portion of the base line between said point of intersection and said second point, means to multiply said seventh and eighth potentials to obtain a ninth potential representative of their product, means to generate a tenth 'potential representative of the square of the actual distance between'said vehicle and said first point, means to generate an eleventh potential representative of the square of the actual distance between said vehicle and said second point, means tovadd said rst, second, and fifth potential to -obtain a twelfth potential representative of the square of the distance between said first point and said vehicle in accordance with said predetermined speed and said heading, means to add said first, sixth and ninth potentials to obtain a thirteenth 14 potential representative of the square of said distance between said second point and said vehicle in accordance with said predetermined speed and said heading, an angle resolver having a stator with two stator windings and a rotor with two rotor windings, means to opposingly impress said tenth potential and said twelfth potential on one of said stator windings,V means to 'opposingly impress said eleventh potential and said thirteenth potential on the other of said stator windings, means to position said rotor at the angle made by said Vehicle heading with said base line whereby there is induced in one of said rotor windings a voltage substantially proportional to the difference between the actual and predetermined speed and there is induced in the other of said rotor windings a Voltage substantially proportional to the deviation of said vehicle from its predetermined heading.

8. Apparatus for obtaining a potential proportional to the product of two functions, each of said functions being expressed as the position of a function shaft comprising a variable frequency oscillator of the type wherein the frequency determining elements consist of a series-connected variable resistor and variable condenser connected in series with a parallel connected variable resistor and variable condenser, said variable resistors being ganged to be simultaneously adjustable and to have equal values, said variable condensers being ganged to be simultaneously adjustable and to have equal values, said variable resistors and variable condensers being calibrated in terms of a respective one of said two functions and respectively mechanically coupled to said function shafts,

l a high impedance connected in series with the output'from said oscillator and a condenser connected in parallel with said oscillator output through said high impedance, the potential output across said latter named condenser being proportional to the product of the impedances to which said variable resistors and said variable condensers are adjusted by said two function shafts.

JOHN B. GEHMAN.

REFERENCES CITED The following references are of record in the file of this patent:

UNITED STATES PATENTS* Number Name Date 2,439,381 Darlington Apr. 13, 1948 2,442,383 Stewart 1 f v June 1, 1948 

